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1.3 Graphical convolution algorithm

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This module discusses the Graphical Convolution Algorithm with the help of examples.

c t ∞ ∞ f g t

Step one

Plot f and g as functions of

Step two

Plot g t by reflecting g over the 'y-axis' ( run time backwards) and then shifting right by t .

Step three

For one value of ' t ' mutiply f g t and compute area underneath the curve to get c t . Area underneath

∞ ∞ f g t c t

Step four

Repeat for all ' t ' to get c t for all t. Usually we will just have to consider several ranges of t.

Step five

Reality check: Does your answer actually make sense?

Remark

Since,

c t ∞ ∞ f g t ∞ ∞ g f t
you can flip and shift either f or g. It is easier to flip and shift the 'simpler' of the two.
Everyone is overwhelmed by convolution at first! Just practise and it will become second nature. Do examples 2.6 to 2.8 in Lathi!

Recall

Now compute output y t for a step input f t u t

Solution

System is LTI with impulse response h t , so use convolution integral y t ∞ ∞ f h t Since, f is simpler, we rewrite it as ∞ ∞ h f t

Step 1

Plot things

Step 2

Do the flip and shift.

Step 3&4

Multiply and integrate.

Case 1

For, t 0

From the fact stated in the caption, ∞ ∞ h f t y t 0 t t 0

Case 2

For t 0

y t ∞ ∞ h f t 0 t 1 R C R C 0 t R C R C R C 1 t R C

Answer

y t 0 t 0 1 t R C t 0

Step 5

Do a reality check: As t tends towhat happens? As t tends towhat happens?

The input is f t t u t and the impulse response is h t 2 t u t . Compute the y t .

Solution

We are given input and impulse response. So ride the convolution convoy! y t ∞ ∞ f h t Both the functions are equally simple, so we flip and shift h t

Case 1

Again y t 0 for all t 0

Case 2

For t 0

y t ∞ ∞ f h t 0 t f h t 0 t 2 t 0 t 2 t 2 t 0 t 2 t 0 t 2 t t 1
y t t 2 t t t 0

Combine case 1 and 2

y t 0 t 0 t 2 t t 0 t 2 t u t

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Source:  OpenStax, My first collection. OpenStax CNX. Aug 03, 2009 Download for free at http://cnx.org/content/col10870/1.1
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